Approximating Early Exercise Boundaries for American Options

American options are different to European style options in that the contract buyer has the right to exercise the option at any time on or before maturity . The freedom to exercise an American option whenever the holder wishes, introduces a boundary problem to solving the Black-Scholes equation popularly used to price the European options. The contract holder will ideally, of course, only exercise the option prior to the expiry date if the present payoff at time t exceeds the discounted expectation of the possible future values of the option from time t to T. So, only if what the holder of the options gets out of exercising early exceeds the market’s view of the expected future return in keeping the option alive, early exercise of the options will take place. Otherwise, he or she will continue to hold on to the option. At every time t there will be a region of values of the underlying whereby it is best to exercise the option (Exercise region) and a complimentary region whereby it is best to keep the option (Free region). There will also be a particular value S(t) of the underlying stock which defines the optimal exercise boundary separating the two regions. In this paper, we evaluate the early exercise boundary using an efficient method developed by Ait Sahlia and Lai (1999) in R and provide numerical results and graphs.